Glasnik Matematicki, Vol. 41, No.2 (2006), 217-221.

ON A FAMILY OF QUARTIC EQUATIONS AND A DIOPHANTINE PROBLEM OF MARTIN GARDNER

P. G. Walsh

Department of Mathematics, University of Ottawa, 585 King Edward St., Ottawa, Ontario, Canada K1N-6N5
e-mail: gwalsh@mathstat.uottawa.ca

Abstract.   Wilhelm Ljunggren proved many fundamental theorems on equations of the form aX2 - bY4 = δ, where δ ∈ {±1, 2, ±4}. Recently, these results have been improved using a number of methods. Remarkably, the equation aX2 - bY4 = -2 remains elusive, as there have been no results in the literature which are comparable to results proved for the other values of δ. In this paper we give a sharp estimate for the number of integer solutions in the particular case that a = 1 and b is of a certain form. As a consequence of this result, we give an elementary solution to a Diophantine problem due to Martin Gardner which was previously solved by Charles Grinstead using Baker's theory.

2000 Mathematics Subject Classification.   11D25.

Key words and phrases.   Diophantine equation, integer point, elliptic curve, Pell equation.

Full text (PDF) (free access)

DOI: 10.3336/gm.41.2.04

References:

  1. J. H. Chen and P. Voutier, Complete solution of the Diophantine equation x2+1=dy4 and a related family of quartic Thue equations, J. Number Theory 62 (1997), 71-99.
    MathSciNet     CrossRef

  2. M. Gardner, On the patterns and the unusual properties of figurate numbers, Sci. Amer. 231 (1974), 116-121.

  3. C. M. Grinstead, On a method of solving a class of Diophantine equations, Math. Comp. 32 (1978), 936-940.
    MathSciNet     CrossRef

  4. D. H. Lehmer, An extended theory of Lucas' functions, Ann. of Math. (2) 31 (1930), 419-448.
    MathSciNet     CrossRef

  5. W. Ljunggren, Zur Theorie der Gleichung x2+1=Dy4, Avh. Norske Vid. Akad. Oslo (1942), 1-27.
    MathSciNet

  6. W. Ljunggren, Ein Satz über die Diophantische Gleichung Ax2-By4=C (C=1,2,4), in: Tolfte Skandinaviska Matematikerkongressen, Lund, 1953, Lunds Universitets Matematiska Inst., Lund, 1954, 188-194.
    MathSciNet

  7. W. Ljunggren, On the diophantine equation Ax4-By2=C (C=1,4), Math. Scand. 21 (1967), 149-158.
    MathSciNet

  8. W. Ljunggren, Collected Papers of Wilhelm Ljunggren, edited by Paulo Ribenboim, Queen's University, Kingston, 2003.
    MathSciNet
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